Approximate and discrete Euclidean vector bundles. (21st March 2023) - Scoccola, Luis; Perea, Jose A.
Abstract: We introduce ε -approximate versions of the notion of a Euclidean vector bundle for ε ≥ 0, which recover the classical notion of a Euclidean vector bundle when ε = 0 . In particular, we study Čech cochains with coefficients in the orthogonal group that satisfy an approximate cocycle condition. We show that ε -approximate vector bundles can be used to represent classical vector bundles when ε> 0 is sufficiently small. We also introduce distances between approximate vector bundles and use them to prove that sufficiently similar approximate vector bundles represent the same classical vector bundle. This gives a way of specifying vector bundles over finite simplicial complexes using a finite amount of data and also allows for some tolerance to noise when working with vector bundles in an applied setting. As an example, we prove a reconstruction theorem for vector bundles from finite samples. We give algorithms for the effective computation of low-dimensional characteristic classes of vector bundles directly from discrete and approximate representations and illustrate the usage of these algorithms with computational examples.